mulclpr - Intuitionistic Logic Explorer

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Theorem mulclpr 7513

Description: Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)

**Assertion**
Ref	Expression
mulclpr	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)

Proof of Theorem mulclpr Dummy variables 𝑞 𝑟 𝑡 𝑢 𝑣 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-imp 7410	. . . 4 ⊢ ·_P = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦 ·_Q 𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦 ·_Q 𝑧))}⟩)
2	1	genpelxp 7452	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ (𝒫 Q × 𝒫 Q))
3		mulclnq 7317	. . . 4 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦 ·_Q 𝑧) ∈ Q)
4	1, 3	genpml 7458	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)))
5	1, 3	genpmu 7459	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))
6	2, 4, 5	jca32 308	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐴 ·_P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))))
7		ltmnqg 7342	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑥 <_Q 𝑦 ↔ (𝑧 ·_Q 𝑥) <_Q (𝑧 ·_Q 𝑦)))
8		mulcomnqg 7324	. . . . 5 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 ·_Q 𝑦) = (𝑦 ·_Q 𝑥))
9		mulnqprl 7509	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑢 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑡 ∈ (1^st ‘𝐵))) ∧ 𝑥 ∈ Q) → (𝑥 <_Q (𝑢 ·_Q 𝑡) → 𝑥 ∈ (1^st ‘(𝐴 ·_P 𝐵))))
10	1, 3, 7, 8, 9	genprndl 7462	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 ·_P 𝐵)))))
11		mulnqpru 7510	. . . . 5 ⊢ ((((𝐴 ∈ P ∧ 𝑢 ∈ (2^nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑡 ∈ (2^nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑢 ·_Q 𝑡) <_Q 𝑥 → 𝑥 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))
12	1, 3, 7, 8, 11	genprndu 7463	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))))
13	10, 12	jca 304	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 ·_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))))
14	1, 3, 7, 8	genpdisj 7464	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))
15		mullocpr 7512	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))))
16	13, 14, 15	3jca 1167	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 ·_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))))
17		elnp1st2nd 7417	. 2 ⊢ ((𝐴 ·_P 𝐵) ∈ P ↔ (((𝐴 ·_P 𝐵) ∈ (𝒫 Q × 𝒫 Q) ∧ (∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ ∃𝑟 ∈ Q 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))) ∧ ((∀𝑞 ∈ Q (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑟 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑟 ∈ (1^st ‘(𝐴 ·_P 𝐵)))) ∧ ∀𝑟 ∈ Q (𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)) ↔ ∃𝑞 ∈ Q (𝑞 <_Q 𝑟 ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))))) ∧ ∀𝑞 ∈ Q ¬ (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∧ 𝑞 ∈ (2^nd ‘(𝐴 ·_P 𝐵))) ∧ ∀𝑞 ∈ Q ∀𝑟 ∈ Q (𝑞 <_Q 𝑟 → (𝑞 ∈ (1^st ‘(𝐴 ·_P 𝐵)) ∨ 𝑟 ∈ (2^nd ‘(𝐴 ·_P 𝐵)))))))
18	6, 16, 17	sylanbrc 414	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 ·_P 𝐵) ∈ P)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 ∧ w3a 968 ∈ wcel 2136 ∀wral 2444 ∃wrex 2445 𝒫 cpw 3559 class class class wbr 3982 × cxp 4602 ‘cfv 5188 (class class class)co 5842 1^st c1st 6106 2^nd c2nd 6107 Qcnq 7221 ·_Q cmq 7224 <_Q cltq 7226 Pcnp 7232 ·_P cmp 7235

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-eprel 4267 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-irdg 6338 df-1o 6384 df-2o 6385 df-oadd 6388 df-omul 6389 df-er 6501 df-ec 6503 df-qs 6507 df-ni 7245 df-pli 7246 df-mi 7247 df-lti 7248 df-plpq 7285 df-mpq 7286 df-enq 7288 df-nqqs 7289 df-plqqs 7290 df-mqqs 7291 df-1nqqs 7292 df-rq 7293 df-ltnqqs 7294 df-enq0 7365 df-nq0 7366 df-0nq0 7367 df-plq0 7368 df-mq0 7369 df-inp 7407 df-imp 7410

This theorem is referenced by: mulnqprlemfl 7516 mulnqprlemfu 7517 mulnqpr 7518 mulassprg 7522 distrlem1prl 7523 distrlem1pru 7524 distrlem4prl 7525 distrlem4pru 7526 distrlem5prl 7527 distrlem5pru 7528 distrprg 7529 1idpr 7533 recexprlemex 7578 ltmprr 7583 mulcmpblnrlemg 7681 mulcmpblnr 7682 mulclsr 7695 mulcomsrg 7698 mulasssrg 7699 distrsrg 7700 m1m1sr 7702 1idsr 7709 00sr 7710 recexgt0sr 7714 mulgt0sr 7719 mulextsr1lem 7721 mulextsr1 7722 recidpirq 7799

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